A shifted cubic ratio sum is at least one at fixed sum three

Proved

WorkbookSource.plus_75202

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that $a+b+c=3$ . Prove that: $\frac{a^3+2}{a^2+8b}+\frac{b^3+2}{b^2+8c}+\frac{c^3+2}{c^2+8a} \geq 1$

Source: InternLM Lean-Workbook, record lean_workbook_plus_75202 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_75202 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a^3 + 2) / (a^2 + 8 * b) + (b^3 + 2) / (b^2 + 8 * c) + (c^3 + 2) / (c^2 + 8 * a) ≥ 1   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_75202; Apache-2.0

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